Advanced Numerical and Experimental Transient Modelling of Water and Gas
Pipeline Flows Incorporating Distributed and Local Effects
YOUNG IL KIM
Thesis for Doctor of Philosophy (PhD)
School of Civil, Environmental and Mining Engineering July 2008
CHAPTER 9
THE EFFECT OF VISCOELASTIC PIPES ON
TRANSIENT PIPE FLOWS
Changes of pipe wall condition may affect the magnitude, phase, and shape of pressure
wave propagation along pipeline systems during a transient event. The geometrical
changes of an inside pipe can be predicted by the proper use of orifice and blockage
models as mentioned in the previous chapter. However, the changes of the pipe material
should be determined by the mechanical behaviour of the material on transient events.
Most transient analysis models use the assumption of linear-elastic behaviour of the pipe
wall. Linear-elastic model is relatively accurate for describing hydraulic transients in
metal or concrete pipes. However, recent tests in water distribution pipe networks show
significant viscoelastic behaviour due to soil-pipe interaction, flexible pipe joints and/or
household water service pipes. In addition, polymer pipes, especially polyethylene pipes,
have been increasingly used in the water and gas industry due to their low price, cost-
effective installation methods and high resistant properties against corrosion. The
viscoelastic behaviour of polymers influences the pressure response during transient events
by attenuating the pressure fluctuations and by increasing the dispersion of the pressure
wave. A mechanical strain principle model (spring-dashpot element model) may be used
to describe this viscoelastic behaviour. The total strain can be decomposed into an
instantaneous and a retarded wall strain. The instantaneous wall strain is analysed by a
linear-elastic model in the basic equations. In this research, a viscoelastic model has been
added to the linear-elastic model in the conservative solution scheme. Unlike the
traditional waterhammer model, the developed transient model is capable of accurately
CH9. The Effect of Viscoelastic Pipes �
292
predicting transient pressure waves both where the entire pipeline is plastic and in a
pipeline with a local plastic section. This chapter focuses on the analysis of hydraulic
transients in a pressurised pipeline system with a local polyethylene pipe component and
presents a number of experimental results illustrating how a localised plastic pipe section
affects the pressure wave in a copper pipe system.
9.1 ADVANTAGES OF POLYETHYLENE PIPE
Polyethylene has grown to become one of the world’s most widely used and recognized
thermoplastic polymer materials, since its discovery in 1933, because of the diversity of its
use [PPI, 2006]. The original application for polyethylene was as a substitute for rubber.
Today’s modern polyethylene resins are highly engineered for much more rigorous
applications such as pressure-rated gas and water pipe. Polyethylene’s use as a piping
material was first developed in the mid 1950’s. Its original use was in oil field production
where a flexible, tough, and lightweight piping product was needed to fulfil the needs of a
rapidly developing oil and gas production industry. The success of polyethylene pipe in
these installations quickly led to its use in natural gas and water distribution systems where
a coilable corrosion-free piping material could be fusion joined in the field to assure a leak-
free method of transporting products to homes and businesses. Now in North America,
nearly 95% of all new gas distribution pipe installation for 12 inch in diameter or smaller
are polyethylene pipe [PPI, 2006]. The performance benefits of polyethylene pipe in oil
and gas related applications have led to its use in equally demanding piping installations
such as water distributions, industrial and mining pipes, marine pipes and other critical
applications where a tough and ductile material is needed to assure long-term performance.
The major reasons for the growth in the use of the plastic pipe, especially polyethylene
pipe, are the cost savings in installation, labor and equipment, the lower maintenance costs,
and increased service life as compared to traditional piping materials. According to the
plastics pipe institute [PPI, 2006], some of the specific benefits of polyethylene pipe (PE)
are introduced in the below.
1) Life Cycle Cost Savings: The life cycle cost of PE pipe can be significantly less
than other pipe materials. The extremely smooth inside surface of PE pipe
maintains its exceptional flow characteristics.
On Transient Pipe Flows
293
2) Leak Free, Fully Restrained Joints: PE heat fusion joining forms leak-free joints
as strong as, or stronger than, the pipe itself. Fused joints can significantly reduce
the potential leak points that exist every 10 to 20 feet when using the bell and
spigot type joints associated with other piping products such as polyvinyl chloride
(PVC) or ductile iron.
3) Corrosion and Chemical Resistance: PE pipe will not rust, rot, pit, corrode, or
tuberculate. It has superb chemical resistance.
4) Fatigue Resistance and Flexibility: PE pipe can be field bent to a radius of 30
times the nominal pipe diameter or less depending on wall thickness. This
eliminates a lot of the fittings otherwise required for directional changes in piping
systems. PE has exceptional fatigue resistance when operating at maximum
pressure. It can withstand multiple surge pressure events up to 100% above its
maximum operating pressure without any negative effect to its long-term
performance capability.
5) Seismic Resistance: The natural flexibility of PE pipe makes it well suited for
installation in dynamic soil environments and in areas prone to earthquakes or other
seismic activity.
6) Construction Advantage: The combination of light weight, flexibility, fully
restrained joints permits considerable time saving and cost-effective installation
methods, such as horizontal directional drilling, pipe bursting, slip-lining, plow and
plant, and submerged or floating pipe. It does not need heavy lifting equipment
because of approximately one-eighth the weight of comparable steel pipe. PE pipe
can be produced over 1,000 feet coiled lengths in certain diameters.
7) Durability: The polyethylene pipe industry estimates a service life for PE pipe to be
50-100 years.
8) Hydraulic Efficiency: The friction factor for other typical pipe materials declines
dramatically over time due to corrosion and tuberculation. However, for water
applications, the PE pipe maintains its smooth interior wall and its flow capabilities
to insure hydraulic efficiency over the intended design life because of their
excellent corrosion and chemical resistance.
9) Temperature Resistance: PE materials retain greater strength at elevated
temperatures or cold weather installations compared to other thermoplastic
materials such as PVC.
CH9. The Effect of Viscoelastic Pipes �
294
9.2 MECHANICAL PROPERTIES OF PLASTIC PIPE
In the case of metal pipes, the conventional tensile test is relied upon to define basic
mechanical properties such as elastic strength, proportional limit and yield strength. These
are important for defining and specifying the pipe material. There are also basic constants
for use in the many design equations that have been developed based upon elastic theory,
where strain is always assumed to be proportional to stress. With plastics there is no such
proportionality. The relationship between stress and strain of plastic material is greatly
influenced by duration of loading and temperature. The stress-stain response shows
hysteresis with the area of the loop being equal to the energy lost during the loading cycle
in Fig. 9.1b, even though near the origin there might appear to be an essentially linear
response. Plastics have no true elastic constants, nor do they have sharply defined yield
points. The values of moduli of plastic materials derived from tensile tests only represent
the initial portion of the stress-strain curve.
stre
ss
strain
stre
ss
strain
stre
ss
strain
stre
ss
strain(a) Purely Elastic Material (b) Viscoelastic Material
Figure 9.1 Stress-Strain Relationship
9.2.1 Viscoelasticity
Plastic pipe is a viscoelastic material that exhibits both viscous and elastic characteristics
due to its molecular nature when undergoing deformation. Unlike purely elastic
substances, a viscoelastic substance is a complex combination of elastic-like and fluid-like
(amorphous) elements that displays properties of crystalline metals and very highly
viscosity fluids. The viscosity of a viscoelastic substance gives the substance a strain rate
that is dependent on time. Purely elastic materials do not dissipate energy when a load is
applied, and then removed. However, viscoelastic materials lose energy during loading
On Transient Pipe Flows
295
and unloading cycle because viscosity is the resistance to activate the deformation.
[Meyers and Chawla, 1998].
The properties of crystalline metals primarily account for the elastic response to stress, in
which elastic materials strain instantaneously when stretched and just as quickly return to
their original state once the stress has been removed (the result of bond stretching along
crystallographic planes in an ordered solid), whereas the amorphous properties of a
vicsoelastic material account for the very high viscosity fluid-like response (the result of
the diffusion of molecules). The overall mechanical response to applied stress is called
viscoelastic since it lies between these two types of behaviour [Ferry, 1970].
The viscoelastic nature of a polymer results in two unique engineering characteristics,
including creep and stress relaxation. These aspects are employed in the design of liquid
piping systems. Creep is the time-dependent viscous flow component of deformation. It
refers to the response of polyethylene, over time, to a constant static load. When a
polymer pipe is subjected to a constant static load, it deforms immediately to a strain
predicted by the stress-strain modulus determined from the tensile stress-strain curve. At
high loads, the material continues to deform at an ever decreasing rate of the load [PPI,
2006]. Stress relaxation is another unique property arising from the viscoelastic nature of
polymer. When subjected to a constant strain that is maintained over time, the deformation
generated by load or stress slowly decreases over time. Stress relaxation describes how
polymers relieve stress under constant strain [Aklonis et al., 1972; Miller, 1996].
Purely elastic materials show an immediate deformation after the application of a sudden
increase in stress and an immediate reformation after removal of the stress. For
viscoelastic samples, this elastic behaviour occurs with a certain time delay. To evaluate
this time-dependent deformation behaviour, two parameters have been defined, relaxation
time and retardation time. Relaxation is the process in a state at rest after a forced
deformation. On the other hand, the term “retardation time” is used for tests when
presetting the stress and when performing creep tests. Retardation is the delayed response
to an applied force or stress and can be described as delay of the elasticity [Mezger and
Westphal, 2006]. The response of viscoelastic pipe systems to loading is time-dependent.
The effective modulus of elasticity can be significantly reduced according to the duration
of the loading because of the creep and stress relaxation characteristics of polymer. The
CH9. The Effect of Viscoelastic Pipes �
296
time-dependent moduli according to retardation time are a key criterion for the design of
polyethylene pipe systems.
9.2.2 Linear Viscoelastic Models
Viscoelastic models have been developed to determine stress-strain interactions and
temporal dependencies of viscoelastic materials. There are two types of viscoelasticity
from the point of modelling. One is linear viscoelastic model when the function is
separable in both creep response and load. It is usually applicable for small deformations.
Another is a nonlinear viscoelastic model where the function is not separable and the
deformations are large. For the large deformations, geometrical non-linearity should be
included. This research focuses on linear viscoelastic models for plastic pipe having
relatively small deformations.
Linear viscoelastic models, which can be expressed by the Maxwell model, Kelvin-Voigt
model, Standard Linear Solid model, and Generalized model [Miller, 1996; Ram, 1997;
McCrum et al., 1997], are based on the mechanical principle associated with
viscoelasticity. For these cases, the viscoelastic behaviour is comprised of elastic and
viscous components modelled as linear combinations of Hookean springs having different
spring constants and Newtonian dashpots containing different viscosity fluids,
respectively. The spring is used to demonstrate ideal elastic behaviour. The deformation
of the spring is directly proportional to the force needed to pull the spring and purely
elastic materials return to their original length when the load is removed as shown in Fig.
9.1a. This relationship is similar to the Hooke’s Law (linear elastic relationship between
stress and strain).
εσ E= (9.1)
where � is the stress, E is the elastic modulus of the material, and � is the strain that occurs
under the given stress. A dashpot can be used to represent the behaviour of viscous
material. When a force is applied to pull the dashpot, the amount of deformation (strain) is
independent of the force but proportional to the velocity at which the force is applied. The
dashpot will not return to its original position once the force is released. The stress-strain
relationship in the dashpot is given as
On Transient Pipe Flows
297
dtdεησ = (9.2)
where � is the viscosity of the material and d�/dt is the time derivative of strain. By
combining various numbers of springs and dashpots, the stress-strain relationships for
different plastics can be approximated. The temperature also affects the strain. For a
given stress level and time, a higher temperature increases the strain.
1) Maxwell Model
Viscoelastic behaviour can be modelled by a purely elastic spring (Hookean spring) and
viscous dashpot (Newtonian dashpot) connected in series known as the Maxwell model as
shown in Fig. 9.5a. The stresses in both elements will be identical. This model represents
a fluid-like material with additional elastic (reversible) deformations. The total
deformation is not reversible. Under an applied axial stress, the total stress and total strain
can be defined as follows
SDTotal
SDTotal
εεεσσσ
+=
==(9.3)
where the subscript D indicates the strain in the dashpot and the subscript S indicates the
strain in the spring. Taking the derivative of strain with respect to time, the relationship
between stress and strain can be defined as follows
dtd
Edtd
dtd
dtd SDTotal σ
ησεεε 1+=+= (9.4)
If a material is put under a constant strain, the stress gradually relaxes. If a material is put
under a constant stress, the strain has two components, elastic and viscous components as
previously mentioned. The Maxwell model predicts that the stress decays exponentially
with time, which is accurate for most polymers. However, it is unable to predict creep in
materials based on a dashpot and a spring connected in series. Although this model is
inadequate for quantitative correlation of polymer properties, it provides a good qualitative
description of linear viscoelastic behaviour [McCrum et al., 1997; Rodriguez, 1982].
CH9. The Effect of Viscoelastic Pipes �
298
The behaviour of this model is considered in creep and stress relaxation. When a fixed
stress �0 is applied to a material at an initial state, we can find the deformation as a
function of time. With constant stress in the spring, �S is constant and d�S /dt = 0.
ησεεε 0=+=
dtd
dtd
dtd SDTotal (9.5)
By integrating Eq. 9.5, we obtain the deformation as function of time.
���
� +=+= −
ησ
ησεε tEtt 1
00
0)( (9.6)
Eq. 9.6 is useful to express results as a time-dependent compliance J(t) = �(t)/�0 = E -1+
t/�. On the other hand, when a fixed deformation �0 is applied at initial state and held, we
can find stress as a function of time.
01 =+=+=dtd
Edtd
dtd
dtd SDTotal σ
ησεεε
(9.7)
By integrating Eq. 9.7, the time-dependent modulus can be defined as
η
εσ /
0
)()( tEEettE −== (9.8)
The term �/E, the relaxation time, is the reciprocal of the rate at which stress decays. The
linear viscoelastic region corresponds to E(t) being independent of �0 [Rodriguez, 1982].
Fig. 9.2 shows the creep and stress relaxation diagrams of Maxwell material model. When
the stress is removed, only the deformation of the spring is recovered. The dashpot
retained permanent deformation.
On Transient Pipe Flows
299
t
�(t)
�0 /E
Slope = � 0 /� Stress removed
Permanent deformation
t
�(t)
�0 /E
Slope = � 0 /� Stress removed
Permanent deformation
t
ln�(
t)
�0=E�0
Slope = -E/�
t
ln�(
t)
�0=E�0
Slope = -E/�
(a) Creep (deformation) (b) Stress Relaxation
Figure 9.2 Creep and Stress Relaxation Diagrams of Maxwell Material Model
2) Kelvin-Voigt Model
Kelvin-Voigt model can be represented by a purely viscous damper and elastic spring
connected in parallel as shown in Fig. 9.5b. This model represents a solid material with
reversible process of deformation when considering a longer duration, rather than a very
short time. The strains in each component are identical and the total stress is the sum of
the stresses in each component because two components are arranged in parallel.
SDTotal
SDTotal
εεεσσσ
==
+=(9.9)
The stress-strain relationship is expressed as a linear first-order differential equation that
can be applied to either to the shear stress or normal stress of a material.
dttdtEt )()()( εηεσ +⋅= (9.10)
If we suddenly supply a constant stress �0 to the viscoelastic material, the deformation
would approach the deformation for a purely elastic material �0/E with the difference
decaying exponentially.
)1()( 0 teE
t λσε −−= (9.11)
CH9. The Effect of Viscoelastic Pipes �
300
where t is the relaxation time and � (=E/�) is the rate of relaxation. Fig. 9.3 shows the
dependence of dimensionless deformation E·�(t)/�0 on the dimensionless time �t. For a
partially liquid viscoelastic material, a certain extent of deformation still remains
permanently even after removing a stress. This value represents the viscous portion. Both
concentrated polymer solutions and polymer melts show this behaviour. The load cycle is
an irreversible process. For a completely solid viscoelastic material, the deformation is
delayed but completely reformed if the period of testing is sufficiently long. The load
cycle is a reversible process because the shape of material will be the same finally when
compared to the initial shape [Mezger and Westphal, 2006].
Dimensionless Time
Dim
ensio
nles
s Def
orm
atio
n
t1*
(1) VE Liquid Partially
(2) VE Solid Completely
Dimensionless Time
Dim
ensio
nles
s Def
orm
atio
n
t1*
(1) VE Liquid Partially
(2) VE Solid Completely
Figure 9.3 Creep and Creep Recovery of Kelvin-Voigt Material
When the solid material is unloaded at time dimensionless time t1*, the elastic element
retards the material back until the deformation becomes zero. The retardation obeys the
following equation.
tettt λεε −=> )()( *1
*1 (9.12)
Although Kelvin-Voigt model is not good at describing the relaxation behaviour after the
stress is removed, it is effective for predicting the creep in the material when contrasted to
the Maxwell model because of reversible process (lim � (t��) = �0/E) of deformation on
long time duration [Meyers and Chawla, 1998].
On Transient Pipe Flows
301
3) Standard Linear Solid Model
This model is the combination of the Maxwell model and Kelvin-Voigt model involving
elements both in series and in parallel as shown in Fig. 9.5c. This alternative is introduced
as the Maxwell model does not describe creep and Kelvin-Voigt model does not
effectively predict stress relaxation. The standard linear solid model is the simplest model
that predicts both creep and stress relaxation. Fig. 9.4 shows the creep response of
standard linear solid model. The process is a combination of creep responses of both the
Maxwell model (shown in Fig. 9.2a) and the Kelvin-Voigt (shown in Fig. 9.3) model.
t
�(t)
�0 /E
Stress removed
t
�(t)
�0 /E
Stress removed
Figure. 9.4 Creep Response of Standard Linear Solid Material Model
The total stress and total strain in the overall system are expressed by Maxwell model as
referred by subscript M.
2
2
1
1
SDM
SDM
SMTotal
SMTotal
εεεσσσ
εεεσσσ
+=
==
==
+=
(9.13)
The stress-strain relationship is expressed as follows
21
12
2
EE
Edtd
EE
dtd
+
���
�−+
=εσση
ηε (9.14)
CH9. The Effect of Viscoelastic Pipes �
302
Although the standard linear solid model can be used to accurately predict the general
shape of the strain curve for viscoelastic materials as well as behaviour for long time and
instantaneous loads, the model shows inaccurate results for strain under specific loading
conditions.
4) Generalized Maxwell or Kelvin-Voigt Model
The Maxwell and Kelvin-Voigt models assume that a single relaxation time and creep
governs the response of the material to a mechanical perturbation. However, there exists
experimental evidence of a distribution of relaxation or creep, which may be considered
either discrete or continuous. It is possible to generalize the Maxwell or Kelvin-Voigt
models so that they more closely represent the actual behaviour of viscoelastic materials
[Riande et al., 2000]. These generalizations can be carried out by assuming an
arrangement of Maxwell elements in parallel or Kelvin-Voigt elements in series as shown
in Figs. 9.5d and 9.5e.
The Maxwell model can be generalized by the concept of a distribution of relaxation times
so that it becomes adequate for quantitative evaluation. This model, also known as the
Maxwell-Weichert model, is the most general form of the models for describing the
viscoelastic behaviour of materials. The generalized Maxwell model expresses the stress
relaxation by using various time distributions. The total time-dependent stress �T(t) is the
sum of the stresses acting on the individual elements �i.
� � −== ii tEiTiT eEt ηεσσ /)( (9.15)
The overall time-dependent modulus is also defined as follows.
�−
== itiEeEttE i
T
TT
η
εσ /)()( (9.16)
The generalized Kelvin-Voigt model is used to analyse the creep and creep recovery
behaviour of solid viscoelastic materials. Each one of the individual Kelvin-Voigt
elements represents the behaviour of individual polymer fraction, which correspondingly
results in an individual retardation time. The total deformation value, by applying the
On Transient Pipe Flows
303
principle of superposition according to Ludwig Boltzmann, is the sum of all individual
deformation values. If the dashpot �0 shown in Fig. 9.5e is eliminated, the model describes
the complete solid viscoelatic materials, such as polyethylene and PVC, and is usually used
for transient analysis for plastic pipeline systems. The details of generalized Kelvin-Voigt
model for pipeline systems are discussed in the section for mathematical model.
E �E �E
�
E
�
(a) Maxwell Model (b) Kelvin-Voigt Model
�
E1
E2 �
E1
E2
Ee
�1
…E1 E2 EN
�2 �N
Ee
�1
…E1 E2 EN
�2 �N
(c) Standard Linear Solid Model (d) Generalized Maxwell Model
E0�1
E1 E2 EN
�2 �N
�0E0�1
E1 E2 EN
�2 �N
�0
(e) Generalized Kelvin-Voigt Model
Figure 9.5 Diagrams of Linear Viscoelastic Models
9.3 VISCOELASTIC BEHAVIOUR ON PIPE FLOWS
Transient analysis model based on linear-elastic behaviour of pipe wall is relatively
accurate to describe transient flows in metal or concrete pipes. However, sometimes
linear-elastic models show significant discrepancies in the magnitude and phase of the
travelling pressure wave for simulating plastic pipes with the property of viscoelastic
behaviour during transient events generated by a rapid change of flow conditions. There
are two different approaches to simulate viscoelastic bahavior of pipeline systems.
CH9. The Effect of Viscoelastic Pipes �
304
One approach is to consider the frequency-dependent factors in actual pipeline systems.
This approach assumes that the effect of the viscoelastic behaviour of the pipe wall can be
described by a frequency-dependent wavespeed. Therefore, a frequency-dependent creep
function is used to calculate the wavespeed during transient events in the frequency
domain. A complex-valued frequency-dependent wavespeed was derived by solving the
wave equation, Eq. 3.36, along with the equation of motion of the rock mass surrounding
the rock-bored tunnel filled with water [Suo and Wylie, 1990a]. The theory of hydraulic
resonance in pipelines is extended to tunnels with frequency-dependent wavespeeds. Their
results show significant changes in resonance conditions by non-linear elastic behaviour of
rock-bored tunnels. Similarly, the frequency-dependent wavespeed in pipeline systems
filled with fluid is considered due to the dynamic effect of the viscoelasticity of the pipe
wall material. Mei�ner and Franke [1977] analysed and compared the damping behaviour
of conduits made of the three different polymers during waterhammer oscillations. They
derived a frequency-dependent wavespeed and the damping factor for an oscillating
pressure wave. In order to take into account the properties of viscoelastic pipes two
frequency-domain methods were developed. Rieutord [1982] used the Laplace-Carson
transform that gave a method that was well suited to the study of laminar transient flows in
linear viscoelastic pipes. The standard impedance or transfer matrix is used to analyse
resonance conditions in periodic pipe flow, and the impulse response method based on the
theory of linear systems is applied to compute non-periodic hydraulic transient pipe flow
[Franke and Seyler, 1983; Suo and Wylie, 1990b].
Another approach is time-domain analysis modelled by analogue spring and dashpot
systems as mentioned in the previous section. In these mechanical principle systems
associated with viscoelasticity, the total strain of a system can be decomposed into an
instantaneous elastic strain and a retarded viscoelastic creep strain. The Kelvin-Voigt or
generalized Kelvin-Voigt model is usually used for analysing the viscoelastic behaviour of
plastic pipe that has the property of solid material viscoelasticity. Williams [1977] found
experimentally that viscoelastic pipes gave rise to strong mechanical damping of the
waterhammer. He measured the transient pressure variation and pipe wall strain in rubber,
PVC, and steel pipes during transient events. Also, he made mention of the expansion
joints in the steel pipe. Rieutord and Blanchard [1979], Ghilardi and Paoletti [1986], and
Ellis [1986] show theoretical studies of the effect of viscoelastic material properties of a
On Transient Pipe Flows
305
plastic pipe based on the generalized Kelvin-Voigt model. Their numerical results showed
the exponential attenuations of wave fronts according to various relaxation times of plastic
pipe. Gally et al. [1979] and Guney [1983] presented a complete mathematical model for
analysing viscoelastic effects on transient pipe flows based on generalized Kelvin-Voigt
model. The calculated results based on the method of characteristics were compared with
experimental results.
Pezzinga and Scandura, [1995] and Pezzinga [2002] presented the results of a theoretical
and experimental study on the use of a high-density polyethylene (HDPE) additional pipe
inserted downstream of a pump in a hydraulic network as a surge suppressor. The
mechanical behaviour of the HDPE was described both by a linear elastic model and by a
Kelvin-Voigt viscoelastic model. Their numerical results showed that the viscoelastic
model better describes the phenomena, but the elastic model adequately estimates the
maximum and minimum oscillations. Covas et al. [2004 and 2005] observed transient data
collected in a high-density polyethylene pipeline system to investigate viscoelastic
behaviour. Their results showed major dissipation and dispersion of pressure waves, and
transient mechanical hysteresis. These measured data were simulated by a generalized
Kelvin-Voigt model, and the creep function of the HDPE pipe was experimentally
determined by creep tests. They noticed that the mechanical tests for the creep function
only provided an estimate of the actual mechanical behaviour of the pipe system when PE
was integrated in a pipeline system, because the creep function depended on not only the
molecular structure of the material and temperature but also on the pipe axial and
circumferential constraints and the stress-time history of the pipeline system. Recent field
tests in water distribution pipe networks showed that concrete, asbestos cement and steel
pipes exhibited some significant viscoelastic behaviour due to soil-pipe interaction,
flexible pipe joints, or household water services [Stephens et al., 2005].
9.4 MATHEMATICAL MODEL
The deformability of the pipe wall is a function of the flow conditions as well as of the
pipe wall characteristics. Pipe distensibility can be a vital part of the transient response for
pressurised liquid pipe flow. The characteristics of pipe deformation are related to the pipe
wall material, cross section geometry and structural constraints. Most transient analysis
models assume linear-elastic behaviour of the pipe wall. Linear-elastic model is relatively
CH9. The Effect of Viscoelastic Pipes �
306
accurate for describing hydraulic transients in metal or concrete pipes. However, transients
in plastic pipes exhibit significant viscoelastic behaviour because these materials have a
different rheological behaviour in comparison to metal and concrete pipes. This section
focuses on the mathematical modelling of hydraulic transients in polyethylene pipes by
adding a retarded strain in the transient pipe flow equations. For linear-elastic behaviour
of the pipe wall, the elasticity of the pipe wall and its rate of deformation are a function of
pressure only as mentioned in Chapter 3. The evaluation of the conduit elasticity is
expressed by A/( pA).
dpeED
AdA = (9.17)
where e is the pipe wall thickness and E is Young’s modulus of elasticity for the pipe wall.
Plastic pipes have both an immediate elastic response and a retarded viscous response.
Therefore, the total strain can be decomposed into an instantaneous elastic strain �e at the
initial state of every process and a retarded strain �r depending on time.
)()()( ttt re εεε += (9.18)
According to the Boltzmann superposition principle, for small strains, a combination of
stresses that act independently in a system results in strains that can be added linearly
[Aklonis et al., 1972; Covas et al., 2005]. The total strain generated by a continuous
application of stress is
( )� ∂∂−+=
tdt
ttJttJtt
0
**
**
0)()()( σσε (9.19)
where J0 is instantaneous elastic creep compliance and J(t*) is the creep compliance
function at time t*. The instantaneous creep compliance J0 is equal to the inverse of
Young’s modulus of elasticity, J0 = 1/E0. According to the following assumptions that i)
the pipe material is homogeneous and isotropic, ii) Poisson’s ratio � is constant for small
strains and a linear viscoelastic behaviour [Gally et al., 1979], the mechanical behaviour of
the deformation of viscoelastic materials is assumed to be only a function of the creep
On Transient Pipe Flows
307
compliance and the total circumferential strain � = (D-D0)/D0 is given directly by the
transposition of the usual strain-stress relationship.
[ ] [ ]� ∂∂−−
−−−+−=
t
dtttJpttp
ttettDttJptp
eDt
0
**
*
0*
*
**
000
00 )()()(2
)()()(2
)( ααε (9.20)
where p(t) is the local pressure, the subscript 0 denotes the initial steady state value, and �
is the parameter function of the pipe constraint [Guney, 1983]. For a pipe anchored at both
ends to eliminate axial strain, � is to be
22
2
2121
�
� � −−++=
De
De
De μμα (9.21)
For a pipe free at both ends
De
De 21 2
2μα ++= (9.22)
where � is Poisson’s ratio. The first and integral term on the right hand side of Eq. 9.20
represent the elastic and retarded strain respectively. To evaluate the time derivative of
retarded strain, it is necessary to know the creep compliance function J(t). The creep
compliance is a function to characterize the time-dependent strain of a viscoelastic material
depending on the molecular structure of the material, temperature and stress-time history.
This function is obtained from dynamic tests carried out on a Rheovibron apparatus using
direct measurements of a sinusoidally varying stress and strain over a restricted frequency
range [Murayama, 1978; Gally et al., 1979]. Nakayasu et al. [1961] presented data on the
viscoelastic behaviour of a HDPE over a wide time scale and a considerable span of
temperature. Schwarzl [1970] proposed a number of approximation formulae for
calculating the storage compliance from creep compliance of viscoelastic materials,
together with bounds for their errors. Short and long term tensile creep tests and their
analysis for obtaining non-linear creep functions of polymers was performed at different
stress levels under various temperature conditions, and the effects of stress and physical
ageing on the creep compliance were studied [Lai and Bakker, 1995; Zhang and Moore,
1997; Mano et al., 2001; Barbero and Julius, 2004; Barbero and Ford, 2004].
CH9. The Effect of Viscoelastic Pipes
GaUy et al. [1979] presented the values of creep compliance and retardation time for
polyethylene at a different temperature as shown in Table 9.1. These values were obtained
by the calculations of the approximation formulae for the creep compliance and the
retardation time of viscoelastic material. Although the creep compliance function can be
obtained by creep or a dynamic test of the material under idealized test conditions, the
mechanical tests for the creep function only provided an estimate of the actual mechanical
behaviour of the material in a real pipeline system because the creep function also depends
on the pipe axial and circumferential constraints, the pipeline support system and the
various stress-time history of the pipeline as a result of flow change.
Table 9.1 Creep Compliances and Retardation Times for Polyethylene
at Different Temperatures [Gaily et at, 1979]
NOTE: This table is included on page 308 of the print copy of the thesis held in the University of Adelaide Library.
The creep compliance function of a pipe wall is described by using the generalized Kelvin
Voigt model for the complete solid viscoelastic materials as shown in Fig. 9.5e after
eliminating the dashpot 110 [Aklonis et aI., 1972].
N
J(t) = Jo+ L/k(1-e-tITk ) (9.23) k=l
where the modulus of elasticity of each spring is Ek = lIJk. The viscosity of each dashpot
is 11k and the associated retardation time is !k = 11kIEk. The parameters Jk and !k of the
viscoelastic mechanical model should be adjusted based on the creep experimental data.
By the combination of the relationship between cross-sectional area and total hoop strain,
dAldt = 2Adddt, and Eq. 9.17, the equation of area state for viscoelastic materials in the
conservative solution scheme is expressed by the following equation.
308
On Transient Pipe Flows
309
tA
tpA
eED
tA r
∂∂+
∂∂=
∂∂ ε2 (9.24)
where the first term on the right hand side corresponds to the elastic strain and the second
term represents retarded strain. The relations between retarded strain �r, its time derivative
��r/�t and pressure are evaluated as the sum of each Kelvin-Voigt element [Gally et al.,
1979; Guney, 1983; Covas et al., 2005].
� � �= =
−
���
���
−==N
k
N
k
t t
k
krkr dteJttxp
eDtxtx k
1 10
*/* *),(
2),(),( τ
ταεε
where )0,(),(),( ** xpttxpttxp −−=−
(9.25)
��== �
��
���
−=∂
∂=∂∂ N
k k
rk
k
kN
k
rkr
txtxpJeD
ttxtx
t 11
),(),(2
),(),(τ
ετ
αεε
where )0,(),(),( xptxptxp −=
(9.26)
The time derivative of retarded strain for each Kelvin-Voigt element is calculated by
analytical differentiation. After mathematical manipulations for each element, it yields the
following numerical first-order approximation [Covas et al., 2005].
( ) ),(~),(
2),(
21
),(2
),(2
),(~
),(~),(
2),(
//
/
ttxet
ttxpeDtxp
eD
eJ
ttxpeDeJtxp
eDJtx
txtxpeDJ
ttx
rktt
kk
tkkrk
k
rk
k
krk
kk
k
Δ−+Δ
Δ−−−−
Δ−−=
−=∂
∂
Δ−Δ−
Δ−
ε
αα
τ
ααε
τεα
τε
ττ
τ (9.27)
After rearranging, the time derivative of total retarded strain is
CH9. The Effect of Viscoelastic Pipes �
310
( )
( )�
�
=
Δ−
Δ−Δ−
Δ−
=
����
�
����
�
�
����
�
����
�
�
Δ−+
���
��� −
Δ−Δ−+
���
���
Δ−−
−=
∂∂=
∂∂
N
K
rkt
tktk
ktk
k
k
k
N
K
rkr
ttxe
et
ettxpJeD
tetxpJ
eD
txpeDJ
ttx
ttx
k
kk
k
1
/
//
/
1
),(~
1),(2
11),(2
1
),(2
),(),(
ε
τα
τα
τ
ατ
εε
τ
ττ
τ (9.28)
This equation is incorporated into the first-order approximated form of Eq. 9.24 to
calculate the retarded stain of viscoelastic material in the conservative solution scheme. In
the following sections, a numerical and experimental investigation is given for the
viscoelastic effects on the rapid transient pipe flows. Each pipe axis is assumed to be fixed
and the pipe wall response is only characterized by the radial distensibility capacity.
9.5 NUMERICAL INVESTIGATION OF VISCOELASTIC PIPE
RESPONSE
Numerical experiments have been undertaken to verify the proposed model and for
investigating the dynamic behaviour of transient pipe flows with polyethylene pipes. The
pipeline system shown in Fig. 9.6 has been used for numerical experiments. This system is
identical with the laboratory pipeline system used for the experimental verification.
Transients are generated by instantaneous valve closure at node 5. The pressure data are
observed at the middle of pipe (node 3) and the downstream valve (node 5).
Tank 1
Valve
Tank 2
node 1 node 2 node 3 node 4 node 5
Tank 1
Valve
Tank 2
node 1 node 2 node 3 node 4 node 5
Figure 9.6 Pipeline System for Numerical Experiments
On Transient Pipe Flows
311
Figs. 9.7 and 9.8 shows the observed pressure waves when the whole pipe material is made
of polyethylene (viscoelastic material) or copper (linear elastic material). The Young’s
modulus of elasticity of polyethylene and copper are 0.649 and 124.1 GPa respectively.
The initial flow velocity is 0.142 m/s with a Reynolds number of 3,662 at 25oC. Fig. 9.7
shows the results of linear elastic model for copper pipe and polyethylene pipe. The results
of polyethylene pipe show a much slower wavespeed and a smaller pressure magnitude
when compared to the results of copper pipe because of the soft pipe material
characteristic.
Observed Data at Node 5
20
40
60
80
0 1 2 3 4Time (s)
Pres
sure
Hea
d (m
)
Copper PipePolyethylene Pipe
Observed Data at Node 3
20
40
60
80
0 1 2 3 4Time (s)
Pres
sure
Hea
d (m
)
Copper Pipe
PolyethylenePi
Figure 9.7 Numerical Results by Linear Elastic Model
Observed Data at Node 5
46
48
50
52
54
56
0 1 2 3 4Time (s)
Pres
sure
Hea
d (m
)
Linear Elastic Model Result
Viscoelastic Model Result
Observed Data at Node 3
46
48
50
52
54
56
0 1 2 3 4Time (s)
Pres
sure
Hea
d (m
)
Linear Elastic Model Result
Viscoelastic Model Result
Figure 9.8 Comparisons between the Results by Linear Elastic and Viscoelastic
Models for Polyethylene Pipe
Fig. 9.8 is the comparison between the results by a linear elastic model and a viscoelastic
model for polyethylene pipe. The values of Table 9.1 are used for creep compliances and
retarded times of the generalized Kelvin-Voigt model. The results of the viscoelastic
model show a significant damping effect of pressure wave magnitude and lagging effect of
CH9. The Effect of Viscoelastic Pipes �
312
the wavespeed when compared to the results of linear elastic model. In particular, the first
pressure rise that is mainly affected by a line-packing effect shows a downward pressure
wave because of the additional distensibility of pipe wall related with viscoelasticity.
The following section shows the analysis of hydraulic transients in pressurised pipeline
system with a local polyethylene pipe section at the middle of the pipeline as shown in Fig.
9.6. The effect of the local viscoelastic pipe is important because the section of existing
old steel and concrete pipes are frequently replaced by polyethylene pipe due to their high
resistant properties against corrosion, low price, and cost-effective installation methods.
The viscoelastic behaviour of a local polymer pipe influences the pressure response during
transient events by attenuating the pressure magnitude and by increasing the dispersion of
the pressure wave. Figs. 9.9 and 9.10 show the simulation results for a linear elastic and a
viscoelastic model at the middle and end of pipe when the copper pipeline has a local
polyethylene section in the middle of pipe. The percentage of each graph presents the
length of local polyethylene section in proportion to the total pipe length.
On Transient Pipe Flows
313
1 % Polyethylene Pipe Section
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
3 % Polyethylene Pipe Section
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1
Time (s)
Pres
sure
Hea
d (m
)
5 % Polyethylene Pipe Section
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
20 % Polyethylene Pipe Section
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
Figure 9.9 Viscoelastic Behaviour of Local Polyethylene Pipe (at the End of Pipe) (gray line: result of whole copper pipeline, blue line: result by linear elastic model,
red line: result by viscoelastic model)
Similar to the effect of an entrapped air pocket on transient pipe flows as mentioned in
Chapter 7, the presence of a local polymer section of even a small percentage has a
significant effect on the character of transients in the copper pipeline. The local
polyethylene sections can either suppress pressure fluctuations or increase the maximum
pressure in pipe. The soft local polyethylene section can be considered as a buffer loaded
with the liquid, so even a small portion of polymer pipe greatly decreases the wavespeed
and pressure magnitude as shown in Figs. 9.9 and 9.10. Also, the graphs show high
frequency excessive pressure spikes caused by interaction between pressure waves and the
polymer section and they can increase the maximum pressure. Unlike the results of whole
polyethylene pipeline system, the simulation results by viscoelastic model for pipeline
system with local polyethylene section show only a slight decrease of wavespeed and
pressure magnitude. Instantaneous linear elastic effect seems to be more dominant
physical phenomenon on transient pipe flow with a local polymer pipe.
CH9. The Effect of Viscoelastic Pipes �
314
1 % Polyethylene Pipe Section
20
40
60
80
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
3 % Polyethylene Pipe Section
20
40
60
80
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)5 % Polyethylene Pipe Section
20
40
60
80
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
20 % Polyethylene Pipe Section
20
40
60
80
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
Figure 9.10 Viscoelastic Behaviour of Local Polyethylene Pipe (at the Middle of Pipe) (gray line: result of whole copper pipeline, blue line: result by linear elastic model,
red line: result by viscoelastic model)
9.6 MEASURED TRANSIENT DATA WITH LOCAL POLYMER
PIPES
Laboratory experiments have been carried out for the verification of the proposed
viscoelastic model and investigation of the real physical phenomena of a local polymer
pipe during transients. The experimental apparatus is described in detail in Chapter 4. The
layout of the pipeline system is repeated in Fig. 9.11.
On Transient Pipe Flows
315
WT ET6106 2787 6013 2755 2905 6015 2895 5975
67144
83135
17872 180081649 (Adaptable Section)
37529 (Unit: mm)
T J1 J2 J3 J6 J7 J8 J9WE EE
WM EM
Brass Block (58 mm)Flow Control Valve (100 mm)
Joint (62 mm)T-junction (94 mm)
WT ET6106 2787 6013 2755 2905 6015 2895 5975
67144
83135
17872 180081649 (Adaptable Section)
37529 (Unit: mm)
T J1 J2 J3 J6 J7 J8 J9WE EE
WM EM
Brass Block (58 mm)Flow Control Valve (100 mm)
Joint (62 mm)T-junction (94 mm)Brass Block (58 mm)Flow Control Valve (100 mm)Flow Control Valve (100 mm)
Joint (62 mm)Joint (62 mm)T-junction (94 mm)T-junction (94 mm)
1649 mm (Long Adaptable Section)
WM EMJ4 J5
702 mm 698 mm
153 mm (Short Section)
J3 J6
48 48
1649 mm (Long Adaptable Section)
WM EMJ4 J5
702 mm 698 mm
153 mm (Short Section)
J3 J6
48 48
Figure 9.11 Pipeline System Layout
Transients are generated at the WE by a side-discharge solenoid valve with a fast operating
time after closing the west flow control valve, thus the pipeline system can be regarded as
a tank-pipe-valve system. The sampling frequency of measured data is 4 kHz. The water
and surrounding temperature is 25oC. The initial steady-state velocities are estimated by
the volumetric method. All transient tests for polymer pipes are undertaken under the
specified 6 different flow condition by adjusting tank pressure as mentioned in previous
chapters. Medium-density polyethylene pipes (PE80B, NP12.5) with three different pipe
lengths are used for local viscoelastic pipe test after insertion into the adaptable section
(between two joints) in Fig. 9.11. The outside diameter is 25 mm and the wall thickness is
2.273 mm. Table 9.2 shows each polyethylene pipe length and the pressure measurement
points.
Table 9.2 Three Different Local Viscoelastic Pipes
Length (m) Location Pressure Measured Points 0.153 J4 – J5 WE, WM, EM, EE 0.895 J3 – J5 WE, EM, EE 1.630 J3 – J6 WE, EE
CH9. The Effect of Viscoelastic Pipes �
316
Figs. 9.12 to 9.17 show the measured transient data at the end of pipeline (WE) under 6
different test conditions used for previous chapters when the pipeline has a section of
polyethylene pipe at the adaptable section. The pressure data with local polyethylene pipe
sections are plotted at the same scale graphs to compare each pressure variation,
wavespeed and pressure wave shape during the same test condition. The wavespeed of the
system is slower when the pipeline has a longer polyethylene section. Also, the section
causes a significant change in the shape and the magnitude of the pressure waves. Similar
to the results shown by the numerical tests, the presence of the local polymer pipe
increases the maximum of surge pressure. The high frequency downward narrow spike of
the first pressure rise, which is caused by a sudden pressure drop when the pressure wave
meets polyethylene pipe section, indicates the location of the section.
These physical interactions between the pressure wave and the local polymer pipe are
much alike with the results of entrapped air pocket on transient pipe flow. However, the
effect of air pocket on transients is less when the initial pressure condition is high because
the initial size of air pocket is more contracted by higher pressure condition, therefore the
actual size of air pocket in the high pressure condition is smaller than that for the low
pressure condition. The pressure wave under the low pressure condition is more deformed
by a relatively large air volume (compare the results of air pocket #4 between Figs. 7.13
and 7.23). On the other hand, the results of a local polymer pipe on transients do not show
the dependency of cavity compressibility according to the initial pressure conditions.
When compared to the results of 1.630 m PE pipe between Figs. 9.12 and 9.17, the shapes
of pressure waves are almost the same because the surface area itself of polymer pipe is an
important physical factor.
On Transient Pipe Flows
317
W it h o u t P E P ip e
- 1
1 0
2 1
0 0 .2 0 .4 0 .6 0 .8 1
0 .1 5 3 m P E P ip e
- 1
1 0
2 1
0 0 .2 0 . 4 0 . 6 0 .8 1
0 .8 9 5 m P E P ip e
- 1
1 0
2 1
0 0 .2 0 .4 0 .6 0 .8 1
1 .6 3 0 m P E P ip e
- 1
1 0
2 1
0 0 .2 0 . 4 0 . 6 0 .8 1
Figure 9.12 Measured Data at the End of Pipeline (WE) under Flow Condition 1
(initial velocity is 0.0599 m/s) (x-axis: measured time (s) and y-axis: pressure head (m))
CH9. The Effect of Viscoelastic Pipes �
318
W it h o u t P E P ip e
0
1 7 .5
3 5
0 0 .2 0 .4 0 .6 0 .8 1
0 .1 5 3 m P E P ip e
0
1 7 . 5
3 5
0 0 .2 0 .4 0 .6 0 .8 1
0 .8 9 5 m P E P ip e
0
1 7 . 5
3 5
0 0 .2 0 .4 0 .6 0 .8 1
1 .6 3 0 m P E P ip e
0
1 7 . 5
3 5
0 0 .2 0 .4 0 .6 0 .8 1
Figure 9.13 Measured Data at the End of Pipeline (WE) under Flow Condition 2
(initial velocity is 0.0824 m/s) (x-axis: measured time (s) and y-axis: pressure head (m))
On Transient Pipe Flows
319
W it h o u t P E P ip e
5
2 7 .5
5 0
0 0 .2 0 .4 0 .6 0 .8 1
0 .1 5 3 m P E P i p e
5
2 7 . 5
5 0
0 0 .2 0 .4 0 .6 0 .8 1
0 .8 9 5 m P E P i p e
5
2 7 . 5
5 0
0 0 .2 0 .4 0 .6 0 .8 1
1 .6 3 0 m P E P ip e
5
2 7 . 5
5 0
0 0 .2 0 .4 0 .6 0 .8 1
Figure 9.14 Measured Data at the End of Pipeline (WE) under Flow Condition 3
(initial velocity is 0.1031 m/s) (x-axis: measured time (s) and y-axis: pressure head (m))
CH9. The Effect of Viscoelastic Pipes �
320
W it h o u t P E P ip e
1 3
3 8
6 3
0 0 .2 0 .4 0 .6 0 .8 1
0 .1 5 3 m P E P ip e
1 3
3 8
6 3
0 0 .2 0 . 4 0 . 6 0 .8 1
0 .8 9 5 m P E P ip e
1 3
3 8
6 3
0 0 .2 0 . 4 0 . 6 0 .8 1
1 .6 3 0 m P E P ip e
1 3
3 8
6 3
0 0 .2 0 . 4 0 . 6 0 .8 1
Figure 9.15 Measured Data at the End of Pipeline (WE) under Flow Condition 4
(initial velocity is 0.1208 m/s) (x-axis: measured time (s) and y-axis: pressure head (m))
On Transient Pipe Flows
321
W it h o u t P E P ip e
2 0
5 0
8 0
0 0 .2 0 .4 0 .6 0 .8 1
0 .1 5 3 m P E P ip e
2 0
5 0
8 0
0 0 .2 0 . 4 0 . 6 0 .8 1
0 .8 9 5 m P E P ip e
2 0
5 0
8 0
0 0 .2 0 . 4 0 . 6 0 .8 1
1 .6 3 0 m P E P ip e
2 0
5 0
8 0
0 0 .2 0 . 4 0 . 6 0 .8 1
Figure 9.16 Measured Data at the End of Pipeline (WE) under Flow Condition 5
(initial velocity is 0.1368 m/s) (x-axis: measured time (s) and y-axis: pressure head (m))
CH9. The Effect of Viscoelastic Pipes �
322
W it h o u t P E P ip e
2 8
6 0
9 2
0 0 .2 0 .4 0 .6 0 .8 1
0 .1 5 3 m P E P ip e
2 8
6 0
9 2
0 0 .2 0 . 4 0 . 6 0 .8 1
0 .8 9 5 m P E P i p e
2 8
6 0
9 2
0 0 .2 0 . 4 0 . 6 0 .8 1
1 .6 3 0 m P E P ip e
2 8
6 0
9 2
0 0 .2 0 . 4 0 . 6 0 .8 1
Figure 9.17 Measured Data at the End of Pipeline (WE) under Flow Condition 6
(initial velocity is 0.1495 m/s) (x-axis: measured time (s) and y-axis: pressure head (m))
On Transient Pipe Flows
323
The relative proportions of local polyethylene pipe in a pipeline system lead to different
patterns of pressure wave and speeds of pressure propagation. Although the whole
polymer pipeline system is beneficial in reducing transient pressure loads, the local
polymer pipe section can cause shock waves (steepening wave front) with high frequency
spikes that significantly increase peak pressures. The formation of shock waves is
associated with the dynamic interaction between fluid flow and pipe wall distensibility.
When the pressure wave meets the local polymer section, it creates high frequency deep
valleys on pressure wave because of the sudden pressure drop due to the instantaneous
expansion of the polymer pipe wall. The polymer has the characteristic of recovery from
the instantaneous expansion and this recovery of the pipe wall compresses pressure wave,
therefore the peak pressure increases.
Fig. 9.18 shows another test example when the pipeline system has a different pipe wall
material at local section. The pressure data are measured at the end of pipeline (WE) and
middle of pipeline (WM) when the pipeline has 0.153 m length rubber (Young’s modulus
E = 0.1 GPa) pipe at the middle of the pipeline (between J4 and J5 in Fig. 9.11). The
specification of local rubber pipe and test condition is the same as the tests for the 0.153 m
polyethylene pipe section mentioned above. The results are similar to the polyethylene
pipe. However, the results show a larger damping rate of the whole pressure magnitude
and a slower wavespeed when compared to the results of polyethylene pipe because the
rubber has a lower modulus of elasticity.
CH9. The Effect of Viscoelastic Pipes �
324
0
10
20
30
40
0 0.2 0.4 0.6 0.8Time (s)
Pres
sure
Hea
d (m
)
Measured data at WE
0
10
20
30
40
0 0.2 0.4 0.6 0.8Time (s)
Pres
sure
Hea
d (m
)
Measured data at WM
(a) Test Condition 2
15
25
35
45
55
65
75
85
0 0.2 0.4 0.6 0.8Time (s)
Pres
sure
Hea
d (m
)
Measured data at WE
15
25
35
45
55
65
75
85
0 0.2 0.4 0.6 0.8Time (s)
Pres
sure
Hea
d (m
)
Measured data at WM
(b) Test Condition 5
Figure 9.18 Measured Data when the Pipeline has a Rubber Pipe Section
9.7 SIMULATION RESULTS FOR EXPERIMENTAL TESTS
Figs. 9.19 to 9.21 show the comparison between measured pressure data and their
simulation results by the proposed viscoelastic model for the polymer pipe section under
the test condition 5 when the pipeline has 0.153, 0.895, and 1.630 m polyethylene pipe
section at the middle of pipeline. The pressure data for the measurement at the end (WE)
of pipeline are shown in Fig. 9.16. The black lines are the measured data, the blue and red
lines are the simulation results by the conservative solution scheme including linear elastic
model and viscoelastic model respectively. The gray lines indicate simulation results when
the pipeline has no polyethylene pipe section (whole copper pipe).
The proposed viscoelastic model, generalized Kelvin-Voigt model, requires a set of creep
compliances (Jk) and retarded times (�k) as input data for existing the polymer pipe section.
These parameters are used to characterize the time-dependent strain of viscoelastic
On Transient Pipe Flows
325
material depending on the molecular structure of the material, temperature, and stress-time
history. Creep or dynamic tests under the idealized test conditions can measure these
parameters experimentally. However, the mechanical test only provides an estimate of the
actual mechanical behaviour of the material in a real pipeline system because the creep
compliance and retarded time also depends on the pipe axial and circumferential
constraints, pipeline support system, and various stress-time history of the pipeline by flow
change as above mentioned.
Alternatively, these parameters can be calibrated by adjusting the measured transient data
to the simulated numerical results [Covas et al., 2002]. The creep compliance and retarded
time may be estimated by minimizing the difference between measured and computed
transient pressure data by an optimisation algorithm. Eq. 9.29 presents the calibrated
initial modulus of elasticity and functions for creep compliance and retardation time of the
tested polyethylene pipe sections for a temperature of 25oC and flow condition 5.
GPaEo 647.3=
27796.0)ln(105545.9)10( 39 +∗⋅= −− τPaJ When � <0.005
25365.0101913.9)10( 39 +∗⋅= −− τPaJ When � >0.005
(9.29)
The results of the whole copper pipe show large discrepancies in both pressure magnitude
and phase with experimental data. The presence of a local polyethylene pipe section has a
significant effect on the character of transients in the copper pipeline. Even small
percentage length (0.153 m polyethylene pipe, 0.4% in proportion to the total pipe length)
of polyethylene pipe causes significant lagging of wavespeed as shown in Fig. 9.21. The
wavespeed of the system is slower when the pipeline has a longer polyethylene section.
Similar to the results of numerical simulation, the results of the viscoelastic model for a
pipeline system with a local polyethylene section show a slight decrease of wavespeed and
pressure magnitude when compared to the results for a linear elastic model. Instantaneous
linear elastic effect seems to be the more dominant physical phenomena than the effect of
viscoelasticity on transient pipe flow with a local polymer pipe section. However, the
results of the viscoelastic model improve the simulation results. The timing of the positive
and negative pressure waves follows the experimental pressure wave quite closely, and the
CH9. The Effect of Viscoelastic Pipes �
326
model accurately predicts the detailed shape of the pressure magnitude affected by local
polyethylene pipe section during a transient event, although the pressure peaks of the
simulation model slightly exceed the measured data. These results indicate that the
pressure transients affected by the local polymer pipe section can be estimated fairly
precisely by the conservative solution scheme including the proposed model.
10
25
40
55
70
85
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
Measured data Whole copper pipeLinear elastic model Viscoelastic model
Figure 9.19 Comparison between Measured and Simulation Data
when the Pipeline has a 1.630 m Polyethylene Section
10
25
40
55
70
85
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
Measured data Whole copper pipeLinear elastic model Viscoelastic model
Figure 9.20 Comparison between Measured and Simulation Data
when the Pipeline has a 0.895 m Polyethylene Section
On Transient Pipe Flows
327
10
25
40
55
70
85
0 0.2 0.4 0.6 0.8 1Time (s)
Pres
sure
Hea
d (m
)
Measured data Whole copper pipeLinear elastic model Viscoelastic model
Figure 9.21 Comparison between Measured and Simulation Data
when the Pipeline has a 0.153 m Polyethylene Section
9.8 SUMMARY AND CONCLUSIONS
Plastic pipe is a viscoelastic material that exhibits both viscous and elastic characteristics
due to its molecular nature when undergoing deformation. Unlike purely elastic
substances, viscoelastic substances are a complex combination of elastic-like and fluid-like
elements that display properties of crystalline metals and very high viscosity fluids. The
viscoelastic behaviour of polymers influences the pressure response during transient events
by attenuating pressure fluctuations and by increasing the dispersion of the pressure wave.
A generalized Kelvin-Voigt model (mechanical strain principle model with spring-dashpot
elements) has been developed for describing the viscoelastic behaviour. Total strain can be
decomposed into instantaneous and retarded wall strain. Instantaneous wall strain is
analysed by linear-elastic model in the basic equations. A viscoelastic model is added to
the linear-elastic model in the conservative solution scheme. The transient model
including the viscoelastic term is capable of accurately predicting transient pressure waves
in both plastic and metal pipeline system with local plastic sections. This research focuses
on the analysis of hydraulic transients in pressurised pipeline system with local
polyethylene pipe.
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Similar to the effect of entrapped air pocket on transient pipe flows, the presence of local
polymer section of even a relatively small size in the copper pipeline greatly decreases the
wavespeed. The relative proportions of local polyethylene pipe in a pipeline system lead
to different patterns of pressure wave and speeds of pressure propagation. Although the
whole polymer pipeline system is beneficial in reducing transient pressure loads, the local
polymer pipe can cause shock waves with high frequency spikes that significantly increase
peak pressures. The formation of shock waves is associated with the dynamic interaction
between fluid flow and pipe wall distensibility. Unlike the simulation results of the pipe
that is entirely polyethylene, the simulation results from a viscoelastic model for the
pipeline system with a local polyethylene section shows a slight decrease of wavespeed
and pressure magnitude when comparing the simulation results to the linear elastic model.
Linear elastic effects are more dominant physical phenomena than the effect of
viscoelasticity on fast transients with short local polymer section. Nevertheless, the results
of viscoelastic model improve simulation results. The timing of the positive and negative
pressure waves follows the experimental pressure wave quite closely and the model
predicts quite well the detailed shape of pressure magnitude affected by local polyethylene
pipe during transients. These results indicate that the pressure transients affected by local
polymer pipe can be precisely estimated by the conservative solution scheme including the
viscoelastic model.
CHAPTER 10
CONCLUSIONS AND RECOMMENDATIONS
The ultimate goal of this research is the development of an appropriate and accurate
transient analysis model for various system conditions to improve the performance of
pipeline condition assessment as proposed in Chapter 1. To achieve this goal, this thesis
presents comprehensive investigations into transient analysis for both water and gas
pipeline systems. The dynamic physical behaviour of various system components, such as
the effects of unsteady wall resistance, viscoelasticity of polymer pipe, local energy loss
elements including leakages, entrapped gas cavities, orifices, and blockages during
unsteady pipe flow conditions have been studied. The dynamic characteristics of these
system components are modelled based on the conservative solution scheme using the
governing equations in their conservative form to improve the accuracy and applicability
of transient analysis in both liquid and gas pipelines. Comprehensive laboratory
experiments have been undertaken for the verification of the proposed liquid and gas
transient analysis models based on the conservative solution scheme and for examining in
significant detail the effect of unsteady pipe wall friction and local loss elements during
transients. These models are useful tools for pipeline condition assessment and fault
detection as well as system modelling and design.
10.1 CONCLUSIONS AND ACHIEVEMENTS
Improvements of transient flow modelling are very important factors for the success of
pipeline condition assessment and fault detection based on transient analysis models that
rely on the accuracy of the model. This research investigates the effects of unsteady wall
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resistance, viscoelasticity of polymer pipe, the dynamic behaviour of local losses including
leaks, entrapped gas cavities, orifices, and blockages on transient pipe flows. The dynamic
characteristics of these system components are modelled based on the conservative
solution scheme to improve accuracy, sensitivity, and applicability of transient analysis in
both water and gas pipeline systems. The following are the major achievements of this
research.
• Development and verification of transient analysis models based on the
conservative solution scheme for water and gas pipeline systems.
• Development and verification of unsteady friction models within the conservative
solution scheme for water and gas pipeline systems.
• Development and verification of leak estimation models for water and gas transient
pipe flows.
• Development and verification of an entrapped gas cavity model for water transient
pipe flow.
• Development and verification of unsteady minor loss models for orifice and
blockage.
• Development and verification of viscoelastic model for plastic pipe, especially for
local plastic pipe section, for water transient pipe flow.
The conservative solution scheme uses fundamental governing equations including all
terms and the subsidiary equations for analysing specified features are also expressed by
the same generic equations. As a result, the developed distributed and local energy loss
models based on the conservative solution scheme are very flexible for the application to
other simplified numerical schemes (for example, MOC) according to the degree of
simplification for reducing the computational time and numerical complexity. The
conservative scheme directly calculates fluid density and pipe wall distensibility at every
computational time step. Thus, the wavespeeds of all nodes are updated at every time step.
This procedure provides big advantages for analysing systems with variable wavespeeds.
The investigations of the unique dynamic behaviour of pressure waves affected by
distributed and local energy loss elements provide fresh insight for pipeline condition
assessment as well as system design. The geometrical changes of pipe wall due to system
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design (valve, connection, or joint) and pipeline fault (corrosion, lining loss, dent, or
partially closed valve) can be predicted by the proper use of the proposed dynamic orifice
and blockage models. Also, the viscoelastic model can be used for simulating soil-pipe
interaction, flexible pipe joint, and household water service that show significant
viscoelastic behaviour as noticed in recent tests in water distribution networks.
Each of the simulation results using four different unsteady friction models (original
weighting function model, approximated weighting function model, and their modified
models for the conservative scheme) in the conservative scheme shows good agreement
with experimental results. The modified model effectively decreases the computational
time. In the case of gas transients, the fluid compressibility is the dominant physical
process for transient analysis rather than the frictional effect. Although the gas transient
models including an unsteady friction term can reasonably accurately predict the whole
transient traces, the simulation results of models overestimate the line-packing.
The leak model for water transients accurately predicts the phase, magnitude and shape of
pressure waves of the experimental data. Unlike the measured data of water transients
with leaks, the change of pressure wave by a leak has been shown to be small for gas
transients. The compressibility of gas diminishes the impact of leaks during transients. A
mathematical model for simulating the effect of entrapped air pockets on transient pipe
flows yields practical results. The gas cavity model is effective in treating relatively small
gas volumes in a pipeline system. This model has been shown to accurately calculate the
overall pressure trace with variable wavespeeds, high frequency pressure spikes, and the
change of pressure magnitude.
The measured transient data with orifices or blockages show that the magnitude of pressure
waves dramatically decreases as the bore size of an orifice or blockage decreases. The
most important characteristics of the measured transient data with restrictions are the
apparent change in the wavespeed illustrated by the lagging and phase change of the
pressure wave due to the reduction of bore size. The simulation results of the frequency
dependent model with wavespeed adjustment for orifices and blockages show good
agreement with the measured data in terms of the magnitude, shape and overall pressure
trace. In addition, the transient model including the viscoelastic term is capable of
accurately predicting transient pressure waves in both polymer pipeline system and metal
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pipeline system with local polymer section.
10.2 RECOMMENDATIONS FOR FUTURE WORK
The main future research is suggested to be the application of the proposed models to field
pipeline systems with large scale and wide range of operational conditions. Real systems
have various product characteristics and pipeline parameters, complex topology, and
numerous flow system components that generate interactions of pressure waves during
transients and offer a challenge to accurate transient analysis. The relatively large
computational time and storage of the conservative solution scheme using implicit finite
difference method can be improved by a more effective sparse matrix solver or advanced
large matrix solver for the problems of large pipeline systems. Moreover, there are still
important and essential local loss elements that need to be investigated to improve the
sensitivity and accuracy of transient analysis and the performance of pipeline
condition/risk assessment and fault detection. Junctions, cross-connections, elbows, and
dead-ends are the most common system components for pipe networks and they may also
create unique dynamic characteristics involving energy dissipation and dispersion during
transient events. Future work will focus on the modelling and investigation of the effects
of these network components on transients.